Sample Problems for Hypothesis Test

Transcription

1 Sample Problems for Hypothesis Test 1. The Bureau of Labor Statistics reported that the average yearly income of dentists in the year 2012 was $110,000. A sample of 81 dentists, which was taken in 2013, showed an average yearly income of $120,000. Assume the standard deviation of the population of dentists s incomes in 2013 is $36,000. a. We want to test to determine if there has been a significant increase in the average yearly income of dentists. Provide the null and the alternative hypotheses. c. Determine the p-value; and at 95% confidence, test the hypotheses. a. H o : $110,000 H a : $110,000 b. Z = 2.5 c. p-value = Since the p-value = , reject H o. Therefore, there has been a significant increase. 2. A tire manufacturer has been producing tires with an average life expectancy of 26,000 miles. Now the company is advertising that its new tires' life expectancy has increased. In order to test the legitimacy of the advertising campaign, an independent testing agency tested a sample of 6 of their tires and has provided the following data. Life Expectancy (In Thousands of Miles) a. Determine the mean and the standard deviation. b. Test to determine whether or not the tire company is using legitimate advertising. Assume the population is normally distributed. Use α = a. = 27, s = 1.67 b. H o : < H a : > Do not reject H o p-value 0.1; do not reject H o and conclude that there is insufficient evidence to support the manufacturer's claim.

2 3. A producer of various kinds of batteries has been producing "D" size batteries with a life expectancy of 87 hours. Due to an improved production process, management believes that there has been an increase in the life expectancy of their "D" size batteries. A sample of 36 batteries showed an average life of 88.5 hours. Assume from past information that it is known that the standard deviation of the population is 9 hours. a. Give the null and the alternative hypotheses. c. At 99% confidence using the critical value approach, test management's belief. d. What is the p-value associated with the sample results? What is your conclusion based on the p-value? a. H o : < 87 H a : > 87 b c. Since Z = 1 < 2.33, do not reject H o and conclude that there is insufficient evidence to support the corporation's claim. d. p-value > ; therefore do not reject H o 4. In the past, the average age of employees of a large corporation has been 40 years. Recently, the company has been hiring older individuals. In order to determine whether there has been an increase in the average age of all the employees, a sample of 64 employees was selected. The average age in the sample was 45 years with a standard deviation of 16 years. Let =.05. a. State the null and the alternative hypotheses. c. Using the p-value approach, test to determine whether or not the mean age of all employees is significantly more than 40 years. a. H 0 : 40 H a : > 40 b. Z = 2.5 c. p-value (.025) < 0.05; reject H 0

3 5. The average gasoline price of one of the major oil companies has been $2.20 per gallon. Because of cost reduction measures, it is believed that there has been a significant reduction in the average price. In order to test this belief, we randomly selected a sample of 36 of the company's gas stations and determined that the average price for the stations in the sample was $2.14. Assume that the standard deviation of the population ( ) is $0.12. a. State the null and the alternative hypotheses. c. What is the p-value associated with the above sample results? d. At 95% confidence, test the company's claim. a. H 0 : 2.20 H a : < 2.20 b. Z = -3 c. p-value = almost zero (0.0013) d. p-value <.05; reject H 0 ; the average price has been reduced. 6. A sample of 81 account balances of a credit company showed an average balance of $1,200 with a standard deviation of $126. a. Formulate the hypotheses that can be used to determine whether the mean of all account balances is significantly different from $1,150. c. Using the p-value approach, what is your conclusion? Let =.05. a. H 0 : = 1150 H a : 1150 b. Z = 3.57 c. p-value (almost zero) <.005; therefore, reject H 0 7. From a population of cans of coffee marked "12 ounces," a sample of 50 cans was selected and the contents of each can were weighed. The sample revealed a mean of 11.8 ounces with a standard deviation of 0.5 ounces. a. Formulate the hypotheses to test to see if the mean of the population is at least 12 ounces. c. Using the p-value approach, what is your conclusion? Let =.05.

4 a. H 0 : 12 H a : < 12 b. Z = c. p-value (.0023) <.005; therefore, reject H 0 8. A lathe is set to cut bars of steel into lengths of 6 centimeters. The lathe is considered to be in perfect adjustment if the average length of the bars it cuts is 6 centimeters. A sample of 121 bars is selected randomly and measured. It is determined that the average length of the bars in the sample is 6.08 centimeters with a standard deviation of 0.44 centimeters. a. Formulate the hypotheses to determine whether or not the lathe is in perfect adjustment. c. Using the p-value approach, what is your conclusion? Let =.05. a. H 0 : = 6 H a : 6 b. Z = 2 c. p-value (.046) ; therefore, reject H 0 9. Ahmadi, Inc. has been manufacturing small automobiles that have averaged 50 miles per gallon of gasoline in highway driving. The company has developed a more efficient engine for its small cars and now advertises that its new small cars average more than 50 miles per gallon in highway driving. An independent testing service road-tested 64 of the automobiles. The sample showed an average of 51.5 miles per gallon with a standard deviation of 4 miles per gallon. a. Formulate the hypotheses to determine whether or not the manufacturer's advertising campaign is legitimate. c. What is the p-value associated with the sample results and what is your conclusion? Let =.05. a. H 0 : 50 H a : > 50 b. z = 3 c. p-value (.0013) is less than.05; reject H 0

5 10. "D" size batteries produced by MNM Corporation have had a life expectancy of 87 hours. Because of an improved production process, it is believed that there has been an increase in the life expectancy of its "D" size batteries. A sample of 36 batteries showed an average life of 88.5 hours. Assume from past information that it is known that the standard deviation of the population is 9 hours. a. Formulate the hypotheses for this problem. c. What is the p-value associated with the sample results? What is your conclusion based on the p-value? Let =.05. a. H 0 : 87 H a : > 87 b. Z = 1 c. p-value = ; therefore, do not reject H At a local university, a sample of 49 evening students was selected in order to determine whether the average age of the evening students is significantly different from 21. The average age of the students in the sample was 23 with a standard deviation of 3.5. a. Formulate the hypotheses for this problem. c. Determine the p-value and test these hypotheses. Let =.05. a. H 0 : = 21 H a : 21 b. z = 4 c. p-value is almost zero; therefore, reject H In order to determine the average price of hotel rooms in Atlanta, a sample of 64 hotels was selected. It was determined that the average price of the rooms in the sample was $ with a standard deviation of $16. a. Formulate the hypotheses to determine whether or not the average room price is significantly different from $112. c. Test the hypotheses. Let = a. H 0 : = 112 H a : 112

6 b. z = c. p-value is 0.08; therefore, do not reject H 0